Existence of nontrivial weak solutions for a quasilinear elliptic systems with concave-convex nonlinearities

Abstract

In this paper, our main purpose is to establish the existence of nontrivial weak solutions to the following systems: $$\left\{ \begin{array}{ll} -\triangle_p u=\lambda V(x)|u|^{r-2}u+F_u(x,u,v),\;\;\; x\in \Omega,\\ -\triangle_p v=\theta V(x)|v|^{r-2}v+F_v(x,u,v),\;\;\; x\in \Omega,\\ u=v=0,\;\;\; x\in \partial\Omega, \end{array} \right.$$ where $\Omega$ is a bounded domain in ${\bf R}^N$, $\lambda,\theta>0$, $\triangle_s u=\mbox{div}(|\nabla u|^{s-2}\nabla u)$ is the s-Laplacian of u. We obtain the existence results in two cases: (i)$1<r<p<N$; (ii)$1<p<r<p^*$. The existence results of solutions are obtained by variational methods.